Reinforced Concrete Design Limit State-Varghese P.C.

Reinforced Concrete Design Limit State-Varghese P.C.

Introduction

1, Methods of Design of Concrete Structures

11 1.2 1.3 1.4 1.5 1.6

17

Introduction 1

Modular Ratio or Working Stress Method (WSM) 2

Load Factor Method (LFM) 2

Limit State Method (LSM) 3

Limit State Method in National Codes "3 Design by Model and Load Tests 4 Publications by Bureau of Indian Standards 4 Review Questions 5 - ¬

Partial Safety Factors in Limit State Design

2.1 2.2

23

2.4 2.5 2.6

27 2.8

Introduction 6

Principles of Limit State Design 6

Procedure for Design for Limit States 7 Characteristic Load and Characteristic Strengths 7

Partial Safety Factors for Loads and Material Strengths 9

Stress-Strain Characteristics of Concrete 11

Stress-Strain Characteristics of Steel 12

Summary of Design by Limit State Method ‘12

Review Questions _ 16 Limit State of Durability of Reinforced Concrete to Environment

3.1 3.2 3.3 3.4 3.5 3.6 3.7

Corrosion of Steel 19

Deterioration of Concrete 20 Prescribed Cover to-Reinforcements 22

Control of Permeability of Concrete 24

Compaction During Placement of Concrete 25 Curing Methods 25

avii xriii

Quality of Aggregates 26 Concrete in Foundations 26

Checking for Limit State of Durability 27 Coated Reinforcements to Prevent Corrosion 27 Determination of Soluble Sulphates 27

Examples 27

Review Questions 29

(Limit State of Collapse—Fiexure) 30-47 4.1 Introduction 30

4.2 — Ultimate Strength of R.C Beams (Lảmit State of Collapse by Flexure) 30 4.3 ‘Balanced, Underreinforced and Overreinforced Sections 31 ~

4.4 Equivalent Compression Block in Concrete 32 :

4.5 — Determination of Constants k¡ and k; for Compression Stress Block 33

4.6 Depth of Neutral Axis of a Given Beam 34 4.7 Importance of Limiting vd Ratios 36 4.8 Calculation of M, by Strain Compatibility Method 36

49 Minimum Depth for a Given M, 39

4.10 Expression for Steel Area for Balanced Singly Reinforced Section , 39

4.12 Expressions for Lever Arm Depth (z) 41 4,13 Calculation of Steel Area for given, b, d and M, for Depths Larger than the Minimum Required 42

4.14 Guidelines for Choosing Width, Depth and Reinforcement of Beams 45

Review Questions 46

Examples in Design and Analysis of Singly Reinforced Beams ——- 48-69 5.1 Introduction 48 -

5.3 Methods of Design and Analysis 48 5.4 Procedure for Analysis of Section by Strain Compatibility (Trial and Error Method)—Method 1 49 Ti

_ 5.7 Necessity for Specifying Maximum and Minimum Tension Steel in.Beams 56

*'5.8 Recommended Procedures for Design and Analysis 57 ,

Examples 58 Review Questions 67

711

7.13 |

714 | T15 7

716 VAT

Desigi

81 2 8.2

83 8.4

85 2

8.6 | 8.7 | 8.8 8.9

8.10;

6.8 Use of Design Aids SP 16: (Method 3) 77 - 6.9 _ Specifications Regarding Spacing of Stirrups in Doubly Reinforced Beams -80 -

Examples 80 Review Questions 87

Problems 87

7.2 Types of Shear Failures 89

7.3 Calculation of Shear Stress 90

7.7 Rules for Minimum Shear Reinforcement 97

7.9 — Step-by-Step Procedure for Design of Links 101

7.11 Enhanced Shear near Supports 103

“7.12 Shearin Slabs 105

7.14 Shear in Members Subjected to Compression and Bending 105

7.15 Shear in Beams of Varying Depth 105

7.16 Detailing of Vertical Stirrups in Wide Beams 107 7.17 Design of Stirrups at Steel Cut-off Points 108

8.2 Effective Flange Width 120

84 T Beam Formulae for Analysis and Design 123

85 Limiting Capacity of T Beams by Use of Design Aids 127 8.6 Expressions for M, and A, for Preliminary Design 128

8.8 Transverse Reinforcement 129 8.9 Tables‘in SP 24 in Design of T Beams 129 8.10 Design of L Beams 129

Problems 145 Design of Bending Members for Serviceability Requirements of Deflection and Cracking 146-169 9.1 Introduction 146

9.2 Design for Limit State of Deflection 147 9.3 Empirical Method of Deflection Control in Beams 147 9.4 Empirical Method of Control of Cracking in Beams 153 9.5 Bar Spacing Rules for Beams 155

9.6 Bar Spacing Rules for Slabs 158

9.8 Curtailment, Anchorage and Lapping of Steel 159 9.9 Stress Level in Steel 159

9.10 Other Requirements 159 9.11 Comments on Minimum Percentages of Steel to be Provided in Beams and Slabs 160

9.12 Recommendations for Choosing Depth of R.C.C Beams 162

10.4 Development Length 172 10.5- End Anchorage of Bars 173

10.9 Equivalent Development Length of Hooks and Bends 177

10.10 Bearing Stresses Inside Hooks (Minimum Radius of Bends) 178 10.11 Anchorage of a Greup of Bars 179

10.12 Splicing of Bars 180 10.13 Lap Splices 180 10.14 Design of Butt Joints in Bars 182 10.15 Welded Lap Joints 183

10.17 Use of SP 16 for Checking Development Length 183 10.18 Importance of Laps and Anchorage Length 184

Review Questions 187 Problems 188

114 11.5 11.6 11.7

Design

12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15) 12.16)

1217 12.18 12.19

Limit

Loadi

13.1 13.2 133) 13.4 13.5 13.6

112 Live Load on Slabs in Buildings 189

113 Structural Analysis of One-Way Slabs with UDL Using Coefficients 191 11.4 Design for Shear in Slabs T91

11.5 Considerations for Design of Slabs 193

Examples 202 Review Questions 212

Problems 213 Design of Two-way Slabs : 214-265 12.1 Introduction 214

12.5 Arrangement of Reinforcements 222 12.6 Negative Moments at Discontinuous Edges’ 223 12.7 Choosing Slab Thickness 223

12.8 Selecting Depth and Breadth of Supporting Beams 224 12.9 - Calculation of Areas of Steel -224-~-~

12.10 - Detailing of Reinforcements 225

12.11 Loads on Supporting Beams 226 12.12 Critical Section for Shear in Slabs 228 12.13 Procedure for Safety Against Excessive Deflection 229

12.14 Procedure for Control of Crack-width 229

12.15" Procedure for Design of Two-way Simply Supported Slabs 229 12.16 Procedure for Design of Two-way Restrained Slabs (with Torsion at Corners) 230

12.17 Concentrated Load on Two-way Slabs 231

12.18 Methods Based on Theory of Plates for Concentrated Loads on Two-way Slabs (Pigeaud’s Method) 231

12.19 Design of Circular Slabs 245

Examples - 250

Review Questions 264 Problems 264

é

Limit State of Collapse in Compression Design of Axially

Loaded Short Columns 266-289 13.1 Introduction 266

13.2 Short Columns 267

1343 Braced and Unbraced Columns 267 - 13⁄4 Unsupported and Effective Length (Height) of Columns 268 13.5 Slenderness Limits for Columns 269

13.6 Derivation of Design Formula for Short Columns 269

X CONTENTS

137 Checking Accidental Eccentricity 271

13.11 Procedure for Design of Centrally Loaded Short Column 274 16 ị 13.12 Strength of Helicaily Reinforced Short Column 276 ị 13.13 Calculation of Spacing of Spirals 277

13.14 Placement of Steel in Circular Columns 278 13.15 Comparison of Tied and Spirally Reinforced Columns 279 13.16 Design of Non-rectangular Columns 279

1317 Detailing of Columns 280 Examples 284

14.3 Uniaxial Bending (Design Assumptions) 291

144 — Stress-Strain Curve for Steel 292

145 Column Interaction Diagram 293 |

147 Application to Circular Sections 297 14.8 Interaction Curves in SP 16 298 14.9 Interaction Diagram forP=0 301 14.10 Shape of Interaction Curves 302 14,11 Accidental Eccentricity in Columns 302 14.12 Use of Interaction Diagrams for Design and Analysis (Method 1) 302 14.13 Design of Eccentric Columns by Equilibrium Equation (Method 2) 303 14.14 Simplified Method—(Method 3) 304

14.16 Simplified BS 8110 Method for Biaxial Bending 308 | 14.17 Shear in Columns Subjected to Moments 309 +ẽ

14:18 Representation of Column Design Charts 309 Examples 310

Design

17.1 17.2 17.3

17.4

17.5 17.6 17.7 17.8 17.9 17.10 17.11 17.12

Design

18.1 18.2 18.3 18.4 18.5 18.6

CONTENTS xi

a, for Specific Conditions as Given in BS 332 Examples 333

Review Questions 339 Problems 340

Introduction 341

Maximum Permitted Length of Columns 342

Basis of Additional Moment Method 342 Expression for Lateral Deflection 342 Reduction Factor for Additional Moment 345

Factors Affecting Behaviour of Slender Columns 349

Design Moment in Braced Columns with Initial Moments 350

Slender Columns Bent About Both Axes 354 Design Procedure 354

Design Procedure to Determine k Values 355

Principle of Moment Magnification Method 355 Examples 357

Design of R.C Walls (According to BS 8110) 375 Design of Plain Walls (According to BS 8110) 376 Design of Transverse Steel in Concrete Walls 377

Rules for Detailing of Steel in Concrete Walls 377

General Considerations in Design of Walls 378 Procedure for Design of Concrete Walls 379

Detailing of Steel 380 : Concentrated Loads on Walls 380

Examples 382

Review Questions 386 Problems 387

Introduction 388 Analysis for Torsional Moment ina Member 389

’ Torsional Shear Stress Analysis of Rectangular Sections 391 Torsional Stress in Flanged Sections 393

Reinforcements for Torsion in R.C Beams 393 Interaction Curves for Combined Shear and Torsion 393

xii CONTENTS | 18.7 Principles of Design of Sections for Torsion by Different Codes: 396 :

18.9 Principles of Design for Combined Bending, Shear and Torsion by IS 456 398

18.10 Detailing of Torsion Steel 400 - :

18.12 Design Procedure ‘According to BS 8110 401 18.13 Arrangement of Links for Torsion in Flanged Beams 403 18.14 Torsion in Beams Curved in Plan 403 \ Examples 404

Review Questions 418 Problems 419

19.1 Introduction 420 19.2 Design Methods for Members in Direct Tension 420 19.3 Elastic Method of Design of Tension Members 421 19.4 Design Procedure for Direct Tension 421

19.5 Design of Members in Bending Tension as in Water Tanks 423 19.6 Minimum Steel Areas and Cover 426

Examples 426 \ Review Questions 428 cnn i

Design of Staircases 431-449 |

20.2 Principles of Design 432 20.3 Applied Loads 434 - Ì 20.4 Design of Stairs Spanning Transversely (Horizontally) 434 - |

1

20.5 Stairs Spanning Longitudinally 434

206 Effective Span 434 20.7 Distribution of Live Loading 435

20.10 Detailing of Steel in Longitudinally Spanning Stairs: 436 20.11 Calculation of Dead Loads and Effective Depths 436

Examples 441

Review Questions 448 Problems 449

21 Design of Corbels, Brackets and Nibs 450-464 | 21.1 Introduction 450

21.2 Allowable Shear in Beams for Lower a,/d Values ị

According to BS 8110 450

21.3 Design ofCorbels or Brackets 451

21.5 Equilibrium of Forces in a Corbel 452 21.6 Analysis of Forces in Corbels 453

ị 21.7 Design Calculation for Steel Areas 456

| 21.8 Procedure for Design of Corbels 457

22.2 Design Loads for Foundation Design 465

22.4 Soil-Pressure on Foundations 466 22.5 Conventional Analysis of Foundations Subjected to

Vertical Load and Moments 466

: 22.6 Design of Independent Footings 468 : 22.7 Minimum Depth and Steel Requirements 471

22.9 Procedure for Design 472

22.10 Design of Square Footings of Uniform Depth 473

ị 22.11 Design of Sloped Square Footings Ã74

22,12.” Detailing of Steel 478

, 22.13 Design of Rectangular Footings 479

431-449 22.14 Plain Concrete Footings 479

22.15 Design of Pedestals 479 22.16 Design of Pile Caps 480 22.17 Under-Reamed Pile Foundations 487 22.18 Combined Footings 491

Examples 492 Review Questions 511 Problems 511

| Appendix A: Working Stress Method of Design 513-529

' A.1 Introduction 513

A.3_ Design Procedure 514

A4 Balanced Sections 517 A.5 Analysis of a Given Section in Bending 520

450-464 A.6 Depth of Neutral Axis with Compression Steel 520

A.7 Other Design Problems by Working Stress Method 520

331

Examples 521

Review Questions 528 Problems 528

Index

BI B.2 B3 B4 B.5 B.6 B7 B.8

xiv CONTENTS

Appendix B: General Data for Designs

Dead Loads 530 Imposed Loads 531 Areas of Bars 532 Areas of Bars at Given Spacings _ 532 Unit Weights and Weights at Specified Spacing of Bars Conversion Factors 533

SI Units 533 Preliminary Estimation of Quantities of Materials 534

Appendix C: Formulae for Some Charts and Tables in IS 456

$33

530-534

535-537 539-541

Fig No |

14 8.8 8.9 8.10 9.5

97 10.6 10.7 10.8 112(a) 11.20) 11.3(a) 11.30)

11.3(¢) 11.34)

1213 | 12.15~12; 12.23 13.6 13.7

149 | 13.10(a)=

1311 7 17.1 17.2

173 18.9 20.5 20.6

20.7 20.8 21.4(a) 214@) 21.6

217

: 9.7 Spacing of side reinforcement 157 , 10.6 Hooks and bend$.in reinforcement bars 177 : 10.7 Laterals at change in direction of bars 180

11.2(a) Layout of steel in one-way simply supported slabs with UDL 196

j Straight bars ¬—

198

200

| 12.15-12.17 Restrained two-way slabs : 240-244

ị 12.23 Arrangement of steel in circular slabs - 249

i 13.6 13.7 Arrangement of steel in tied columns 273

Arrangement of steel in spirally reinforced-columns 277

Ị 13.9 Splicing of columns 280

18.9 20.5 Reinforcing flanged sections in torsion 403

Reinforcement drawing of staircase slabs continuous with landing 438

439

454

21.4(b) Reinforcement drawirig of corbel ‘when using 16 mm diameter or less

| 21.6 21.7 Reinforcements in nibs with large loads , 460

22.6 22.8

xvi List oF ILLUSTRATIONS SHOWING DETAILING OF REINFORCEMENTS

Layout of steel for R.C pedestal 480 ,Reinforcement drawing of pile caps 486

*? Arrangement of reinforcément for pile caps 487 \ Typical details of bored cast in situ under-reamed pile foundations 488 |

3.1 | 3.4(a)

10.1

43 4.4 4.5 4.6 3.1 5.2 5.3 6.1 6.2

63 6.4

91 9.2

93 10.1

Caption Factored loads for limit state design

Partial safety factor for strength, 7„

Treatment of concrete exposed to sulphates _ Classification of exposure conditions

Nominal cover for reinforcement for mild conditions of exposure Classification of exposure conditions according to IS 456 Nominal cover for durability

Increased.cover for special conditions for concrete below M25 Minimum cement content and water-cement ratio for durability

Limiting values of x/d

Values of constants for maximum compression block Table of values of resistance moment for limiting values of x/d Percentage of limiting steel areas (p,) for balanced design Values for lever arm depth-factor ~~~

Minimum’ beam widths (mm) for reinforcements

Flexure-reinforcement percentage, p,, for singly reinforced sections (f, = 15 N/mm?)

Flexure-reinforcement percentage, p,, for singly reinforced sections (f,, = 20 N/mm?)

Stress in compression reinforcement for d’/d ratios

Salient points on design stress-strain curve Flexure-reinforcement percentages for doubly reinforced

Flexure-reinforcement percentages for doubly reinforced

sections (f,, = 20, f, = 415)

Design shear strength of concrete, t,, N/mm?

Maximum shear stress in concrete, 7 max

Shear-vertical stirrups

Design of beams for shear A,,/s, values for different values of s, and 6 Shear—bent-up bars

Procedure for T-beam design ' Basic values of span-effective depth ratios for deflection control of beams

Classification of structures according to crack width

Clear distance between bars (mm) Design anchorage.bond strength of deformed bars (ta)

174 17.5

“176 18.1 18.2 18.3 19.1 19.2 19.2(a) 19.3 19.4 19.5 19.6 22.1 22.2

AI A2 A3

AA

BI B.2 B43 B4

Caption

Specifications and equivalent lengths of bends and hooks

Bending moment and shear force coefficients for beams.and oné-way slabs Shear multiplying factor for slabs

Spacing of distribution steel for slabs (cm) Values of k for concentrated loads on slabs (one-way slabs) for

equivalent width method Bending moment coefficients for simply supported two-way slabs Bending moment coefficients for rectangular panels supported on four sides with provision for torsion at corners - Shear force coefficients for uniformly loaded two-way rectangular slabs

Modification of Pigeaud’s method for eccentric loads Bending moment reduction factors for continuity in Pigeaud’s method’

Moments and shears in circular slabs with uniform load = w/m”

Values of design stress-strain curve for Fe415 steel Stress block parameters when the neutral axis lies outside the section Effective length coefficients

Values of k, and k, for values of P, Effective height of unbraced plain concrete walls

Effective height of braced plain concrete walls Slenderness limits of concrete walls Influenee of height-length ratio on strength of R.C walls -

Stress-reduction factor for plain concrete walls (a) Minimum reinforcement in walls

Values of K and @ in torsion of rectangles Design for shear and torsion BS 8110 (1985) Moment coefficient for torsion in ring beams

Allowable stresses in steel for direct tension Allowable stress in concrete in direct tension without cracking of concrete

Allowable stresses in concrete in direct tension allowing cracking

of concrete Tension lap lengths Permissible concrete stresses for strength calculation by elastic method (N/mm?)

Permissible steel stresses for strength calculation by elastic method Permissible concrete tension in bending

Safe load for vértical under-reamed piles Recommended sizes of beams over under-reamed piles

Permissible stresses in concrete

Permissible stresses in steel reinforcement

Beam factors by W.S method

Percentage of tensile reinforcement for balanced section Dead loads

Imposed loads Areas of bars (nm)

Areas of bars at given spacings (mm?)

Preliminary estimation of quantities of materials

List of TasLes xix

Each of these may be subjected to various combinations of forces with the material itself undergoing

effects of creep, shrinkage, temperature variations as well as environmental influences that affect the

durability of the structure

Design of a reinforced concrete structure is carried in many stages, for instance, the empirical apportionment of economical sizes to the various elements, the detailed calculation of the strength

and stability of the structure as a whole, and each of the elements under the various forces it is subjected to, the estimation of the economical amount of reinforcements to be provided for safety,

as also the detailing of the steel in various parts for integrated action In addition, serviceability aspects (e.g deflection and cracking) and durability aspects (e.g corrosion and deterioration of

concrete) should also be given due consideration in the design

Starting from a purely empirical approach adopted at the tum of this century, reinforced concrete construction has undergone a phase of apparently rigorous elastic theory Since then: we have realised that the semi-empirical approach as advocated by Limit State Design is the best method for design

of concrete structures Thus, after the CEB-FIP recommendations on Limit State Design were published

in 1970, Limit State Design approach has been adopted internationally, in the USA by ACI-318-71,

in the UK by CP 110-1972, in Australia by AS 1480-1974, and in India by IS-456-78 It should, however, be noted that even though the various aspects ‘of R.C design are controlled by these codes and regulations, the structural engineer must exercise caution and use his judgement in addition to calculations in the interpretation of the various provisions of the code to obtain an efficient and

economical structure Associated design charts and tables offer great help to shorten the lengthy

calculations required Reference to more than one code brings in a deeper insight into the current

state of knowledge on the subject Besides, detailing of reinforcement, which is an art, has to be

carried out according to the recommendations given in approved manuals,

It is evident from the foregoing’ discussion that to write a book incorporating all the above components of an efficient reinforced concrete design is not an easy task Yet, Prof P.C Varghese has been able to bring out a text book by combining admirably all thesé elements I consider this book to be one of the most comprehensive and yet simple text books that has been published so far

in India on the subject

Professor Varghese had himself gained knowledge of Reinforced Concrete Design from his teachers

at Harvard University, and Imperial College, London His subsequent teaching and research career

at the Indian Institute of Technology (IIT) Kharagpur and IIT Madras lasting over two decades

during the time R.C design was being revolutionised has reinforced his knowledge on the subject

As UNESCO Technical Adviser at the University of Moratuwa, Sri Lanka, he had the opportunity

to introduce Limit State Design based on the British code No wonder then, he has been able to

integrate the best Indian, British and American practices in this text In addition to explaining the theoretical aspects of the design calculations, he has worked out adequate number of examples to:

bring out the salient features of R.C design It will be advisable if educational institutions inculcate

xxi

xxii Foreworp

in the students the habit of working out design problems in the professional format presented in the

book from the beginning The review questions given at the end of each chapter will ensure, if answered completely by the students, a thorough comprehension of the subject

This book should prove to be an ideal text book for the students, as well as an able companion for teachers and those interested in updating their knowledge and expertise on the subject

It is an honour for anyone to write the Foreword for such a commendable text book, and this is particularly so to me who has been an ardent follower of Prof P.C Varghese for the past many years

in his different activities—teaching, research and consultancy

P Purushothaman Fomerly Professor of Structural Engineering

and Dean, P.G Studies

College of Engineering, Guindy Anna University, Madras

Method h

Limit Stat

As mat mandatory should be Ultimate

states suc!

Limit Stai

This be Indian St2

Method is a sheer waste of time and effort Such teaching also creates confusion in the minds of the

average students A design-oriented subject should be taught as it is ptofessionally practised

After the publication of the International Recommendations for the Design and Construction of

Concrete Structures by CEB-FIP in 1970, the whole world.has accepted the principles of Limit State

Design for design of concrete structures in their various national codes The Uriited Kingdom (UK)

was the first country to comipletely switch over to the new design practice by replacing CP 114 (1969) by CP 110 (1972), which was again revised as the present code of practice, BS 8110 (1985)

It deals with both reinforced concrete and prestressed concrete structures, The first Indian code, the Code of Practice for Plain and Reinforced Concrete, was published in

1953 It was revised first in 1957 and-then in 1964 under the title “Code of Practice for Plain and

‘Reinforced Concrete” In 1978 India also accepted the recommendations of CEB-FIP and published

the present code IS 456 (1978) As both the British and Indian Codes follow the recommendations

of CEB-FIP, many of the ideas of the two codes-are-similar iii nature 18-456 (1978) retains its title

as “Code of Practice for Plain and Reinforced Concrete”, and a separate code IS 1343 (1980) deals with design of Prestressed Concrete IS 456 (1978) is divided into six sections, with the first five

sections written along the Limit State Design principles, and the last section on the Working Stress

Method has been retained as an alternative method of design so that a gradual changeover to the Limit State Method can take place in the profession

As many years have already elapsed since the publication of IS 456 (1978), most of the practising engineers in India have already adopted the new method of design and it has obviously become mandatory for the educational institutions also to switch over to teaching the Limit State Design It

should be pointed out that (as explained in detail in the book), Limit State Design is not simply Ultimate Load Design, which is only one of the limit states to be considered Many additional limit

states such as deflection, cracking and durability have to be accounted for in the total design by the Limit State Method

This book is not written to replace the code and the other valuable publications of the Bureau of

Indian Standards, It is meant only to explain the provisions of these publications from fundamentals

and make the publication more familiar to the students, Hence to get the maximum benefit, this book

has to be used along with the following publications of the Bureau of Indian Standards:

1 IS 456 (1978) Code of Practice for Plain and Reinforced Concrete

2 SP 16 (1980) Design Aids to IS 456 (1978)

3 SP 34 (1987) Handbook on Concrete Reinforcement and Detailing

All students should buy copies of these very useful documents Only sample charts, tables, figures, etc have been reproduced here with permission from BIS to illustrate the use of these publications

for design Many definitions, list‘of symbols etc have been purposely omitted in this text Students

Should consult the BIS publications in this regard In general, easily understandable internationally

Xxiv PREFACE

accepted symbols are used throughout the book The readers are advised to refer to SP-24

—Explanatory Handbook on IS 456 for a better understanding of the various provisions of the code

As this text is the outcome of the lectures I delivered for several years for the first compulsory course on Reinforced Concrete Design, it does not deal with all the provisions of IS 456 (1978), but

with only those topics that all civil engineers should know A second course covering advanced

topics such as deflection, crack width, flat slabs, deep beams, ribbed floors, beam column connections

etc are offered as an elective to selected undergraduate students or as a basic course to all postgraduate

students in Structural Engineering It is hoped that these will be published as a separate volume at

a later date

The text has been class tested and was well received by many batches of my students I fervently

subject

This year, the College of Engineering, Guindy, Anna University—an institution which has been

in the vanguard of education in engineering and technology—is celebrating its two hundredth year

of existence As an Honorary Professor of the University, I have great pleasure in presenting this

book during the bicentennial celebration of this great institution

P.C Varghese

lề Ye

I wish to acki and publicati

I wish to acknowledge the help received from various individuals and institutions during the preparation

1 studied the fundamentals of modern Reinforced Concrete Design first under Prof Dean Peabody,

Jr at Harvard University and then undér Prof A.L.L Baker at Imperial College, London To both

of them I am indebted for creating in me an interest in the subject,

It was during my teaching career at the Indian Institute of Technology (IIT) Kharagpur and then

at IIT Madras, lasting over twenty years, that I took up postgraduate teaching and research in

Reinforced Concrete While I was working as UNESCO Technical Adviser at the University of Moratuwa, Sri Lanka, I got the opportunity to teach Reinforced Concrete Design based on the British Code on Limit State Design for nearly ten years To these institutions and the students I taught there,

T owe a debt of gratitude for the help I received to evolve this textbook from the lectures delivered

1 am indebted to Prof V.C, Kulandaiswamy, former Vice-Chancellor, Anna University, Madras

for his invitation to work with the University as Honorary Professor after my retirement and to Prof M Anandakrishnan, the present Vice-Chancellor for his encouragement to.continue my association

with the University Also, Professor P._Purushothaniaint of Anna University has rendered valuable help by reading as well as correcting the manuscript and using it in his classes at the University

Suresh Mathen and M.A Abraham have contributed by checking the examples given in the text

‘ while they were working with me as engineer-trainees

Acknowledgement is also due to the Bureau of Indian Standards for liberally granting permission

to reproduce in this book typical tables, charts; figures and other materials from their publications,

IS 456 (1978), SP 16, SP 24, and SP 34 It is hoped that explanations and illustrations of the use

of.these very useful publications in this book will lead to their wider use by the students and designers in India :

Finally, I wish to put on record my appreciation for the excellent cooperation received from the

Publishers, Prentice-Hall of India, New Delhi, both during editorial and production stages

of elasticity of concrete, and permissible working stresses A start on the recent development leading

to limit state design, otherwise called strength and: performance criterion, can be said to have been

made from the date of creation of.the European Committee for Concrete (Comite European du

Beton) called.CEB, in 1953 The initiative for this came from the reinforced concrete contractors of France The Committee has its headquarters at Luxembourg Its objectives are the coordination and synthesis of research on safety, durability and design calculation procedures, for practical application

to construction Their first recommendations for reinforced concrete design were published in 1964

Later, under the leadership of Yves Guyon (well known for his expertise on prestressed concrete), the CEB established technical collaboration with the International Federation for prestressing

(Federation International de la Preconstrainte), called FIP Recommendations for international adoption for design and construction of concrete structures were published by them in June 1970 and the

“CEB-FIP Modei Code for Concrete Structures” was proposed-in 1977 These efforts formed the solid bases for the creation of an “International Code of Practice”: Through these publications a

unified code for design of both reinforced and prestressed concrete structures was developed

According to the above model code, structural analyses, for determination of bending moments, shears etc are to be carried by elastic analysis, but the final design of the concrete structures is to

be done by the principles of limit state theory

The model code was to be a model from which each country was to write its national code, based

on its stage of development but agreeing on important points, like method of design for bending, shear, torsion etc., to the model code The basis had to be scientifically rigorous, but compromises could be made because of inadequacy of data on the subject for any region

The British were the first to bring out a code based on limit state approach as recommended by the CEB-FIP in 1970 This code was published as Unified Code for structural concrete (CP 110:

1972) Other countries in Europe and the United States adopted similar codes, and today most countries follow codes baséd on the principles ‘of Limit State Design

India followed suit during the revision of IS code 456 in 1978, and the provisions of the limit state design (as regards concrete strength, durability and detailing) were incorporated in the revised

code IS 456 (1978) in Sections 1-4 However, for design calculations to assess the strength of an

R.C member, the choice of either limit state method or working stress method has been left to the designer (Sections 5 and 6) with the hope that with time, thé working stress method will be completely

teplaced by the limit state method Many of the Provisions of IS code are very similar to the BS approach

A uniform approach to design, with reference to the various criteria, is the dream of all reinforced

concrete designers with an international outlook, but it is bound to take many more years to come

into effect In the USA the code used for general design of reinforced concrete structures is the

“Building Code Requirement for Reinforced Concrete” ACI 318 (1983) The general principles of

Seren

xxviii INTRODUCTION

limit state design are named as strength and serviceability method in the above code It is also interesting

to note that among the European Common Market countries there is a move to unify the codes of

the various member countries

As résearch in various aspects of concrete design is still being carried out in many countries and

these countries are anxious that the results of these latest research are reflected in their national

codes, it will take a long time for all the codes in the world to be the same It is therefore advisable

that a student be aware of at least the general provisions of the codes of other countries too It is

for this purpose that at many places in this book, IS, BS and ACI provisions are briefly discussed

and- compared

As has happened in other scientific fields, new ways of thinking replace the old ways In scientific

circles this is generally referred to as a paradigm shift Limit state design should therefore be looked

upon as a “paradigm”, a better way of explaining certain aspects of reality and a new way of thinking

about old problems Thus, it should be learned and taught with its own philosophy, and not as an

extension of the-old elastic theory This.book is therefore exclusively devoted to the study of ‘Limit

State Philosophy’ and is written with the hope that it will give the reader insight into the philosophy

of Limit State Method for design of concrete structures

the natiof

The Indian St follow th its revise

German

The methods

Reinforced concrete members are allowed to be designed according to existing codes of practice

by one of the following two methods (IS 456: clauses 18.2 and 18.3):

1 The method of theoretical calculations using accepted procedures of calculations

2 The method of experimental investigations

The theoretical methods are employed for design of the commonly used structures These, methods

consist of numerical calculations based on the procedures prescribed in codes of practices prevailing

in the country Such procedures are based on one of the following methods of design:

1 The modular.ratio or the working stress method, also known as the elastic method

3 The limit state method

The experimental methods are used only for unusual structures and are to be carried out in a properly equipped laboratory by (a) tests on scaled models according to model analysis procedures,

and (b) tests on prototype of the structure This book deals only with the methods based on

theoretical calculations, and hence reference should be made to other published literature for methods

of design by experimental investigations As already mentioned, experimental methods arise only

when one has to deal with unusual structures about which sufficient data on theoretical methods

of calculations is not available

The theoretical methods themselves are the result of extensive laboratory tests and field investigations Safe and universally accepted methods of calculations based on strength of materials

and applied mechanics have been derived from these laboratory investigatiofis and are codified into

the national codes The code of practice to be used in India at present is the one published by the Bureau of

Indian Standards IS 456-1978 All reinforced concrete structures built in India are required to follow the provisions of these codes IS 456 is very similar to the British Codes CP 110 (1973) and its revised version BS 8110 (1985) The American practice follows the ACI Code 318 (1983), the

German practice, DIN 1045, and the Australian practice, AS 1480

The Indian Code at present allows the use of both the working stress and the limit ‘state methods of design However, as more and more countries are adopting only the limit state method -

of design, we can expect that India will also, in the near future, discard the modular ratio method

and follow the limit state method ‘for design of reinforced concrete’ structures

This chapter deals briefly with the various theoretical: methods of design mentioned above

2, LIMIT STATE DESIGN OF REINFORCED CONCRETE

1.2 MODULAR RATIO OR WORKING STRESS METHOD (WSM)

This method of design was evolved arouind-1900 and was the first theoretical method accepted by

national codes of practice for design of reinforced concrete sections It assumes that both steel and

moduli of elasticity of steel and concrete) can be used to determine the stresses in steel and

concrete This method adopts permissible stresses which are obtained by applying specific factors

of safety on material strength for design It uses a factor of safety of about 3 with respect to cube

strength for concrete and a factor of safety of about 1.8 (with respect to yield strength) for steel

Even though structures designed by this method have been performing their functions satisfactorily for many years, it has three major defects First, since the method deals only with the

elastic behaviour of the member, it neither shows its real strength nor gives the true factor of safety

of the structure against failure Second, modular ratio design results in larger percentages of compression

steels than is the case while using limit state design, thus leading to uneconomic sections while

dealing with compression members or when compression steel is used in bending members Third,

the modular ratio itself is an imaginary quantity Because of creep and nonlinear stress-strain

relationship, concrete does not have a definite modulus of elasticity as in steel

In the modular ratio method of design, the design moments and shears in the structure are calculated by elastic analysis with the characteristic loads (service loads) applied to the structure;

the stresses in concrete and steel in the sections are calculated on the basis of elastic behaviour of

the composite section An imaginary modular ratio which may be either a constant in value for all

strengths of concrete or one which varies with the strength of concrete is used for calculation’ of

the probable stresses in concrete and steel =

CP 114, the code used in U.LK till 1973, recommended the use 2 of a constant modular ratio

of 15, independent of the strength of concrete and steel Other codes ‘like IS 456 recommend a

modulus of elasticity of concrete which varies with the strength of concrete

It should, however, be noted that modular ratio method with due allowance for change of the

value of modulus of concrete to.allow for creep, shrinkage etc is the only method available when

one has to investigate the R.C section for service stresses and for the serviceability states of

deflection and cracking Hence a knowledge of working stress method is essential for the concrete

designer and forms part of limit state design for a serviceability condition This method is explained

in detail in Appendix A at the end of the text :

13 LOAD FACTOR METHOD (LFM)

A major defect of the modular ratio method of design is that it does not give a true factor of safety

against failure To overcome this, the ultimate load method of design was introduced in R.C

design This method, later modified as the Load Factor Method (LFM), was introduced in U.S.A

in 1956, in U.K in 1957, and later on in India In this method, the strength of the R.C section at

working load is estimated from the ultimate strength of the section The concept of load factor,

which is defined as the ratio of the ultimate load the section can carry to the working load it has

" _ to carry, was also introduced in U.K Usually, R.C structures are designed for suitable separate

load factors for dead loads and for live loads with additional safety factors for strength of concrete

After the introduction of the load factor method, in order to make the calculations comparable

“with the modular ratio method, sdme codes like the British and the Indian codes adopted the

Modified Load Factor Method This method used the ultimate load principles for design, but retained

the allowable service stresses concept in the calculations Thus CP 114 (the code used earlier in

concrete act together and are perfectly elastic at all stages so that the modular ratio (ratio between

of concrete

concrete sty place due t

many of the were evolvs

Concrete” p

the Blue Be Concrete St

for Prestres: the “Model codes have

In thị

ACI 318-83

design The] design, and

METHODS OF DESIGN OF CONCRETE STRUCTURES 3

U.K.) used a load factor (ratio of ultimate load to working load) of 2 with additional safety factor

applied to material strength, to arrive at the permissible service stresses As the variation of strength

of concrete is much more than that in steel, an additional factor of safety of 1.5 (i.e, 3/2) for designed mixes and 1.67 (i.e 5/3) for nominal mixes were used when calculating the permissible concrete stresses This additional factor of:safety for concrete also ensured that failure always took place due to tension failure of steel, and not by sudden compression failure of concrete It should

be noted that historically the load factor method was the first method which did not use the

imaginary modular ratio for design of reinforced concrete members As this method has since been

superseded by the limit state method in codes of practice, today it is not.necessary for the student

to make a separate study of the load factor method of design in great detail

1.4 LIMIT STATE METHOD (LSM) Even though the load factor method based on ultimate load theory at first tended to discredit the traditional elastic approach to design, the engineering profession did not take to such design very readily Also, steadily increasing knowledge brought the merits of both elastic and ultimate theories into perspective It has been shown that whereas ultimate theory gives a good idea of the strength

aspect, the serviceability limit states are better shown: by the elastic theory only

Since a rational approach to design of reinforced concrete did not mean simply adopting

the existing elastic and ultimate load theories, new concepts with a semi-probabilistic approach

to design were found necessary The proposed new method had to provide a framework which

would allow designs to be economical and safe ‘This-new- philosophy of design was called the

Limit State Méthod (LSM) of design It has been already adopted by many of the leading

“countries of the world in their codes as the only acceptable method of design of reinforced concrete

Provisions in both the Indian and the British Codes for limit design are very similar, and many of the coefficients and tables recommended for design have the same value Both of them

were evolved from the “Recommendations for an International Code of Practice for Reinforced - Concrete” published by CEB (the European Committee for Concrete) in 1963, generally known as

the Blue Book, and the complementary report “International recommendation for the Design of Concrete Structures” published in 1970 by the CEB along with FIP [The International Federation

for Prestressing], commonly known as the Red Book These were revised in 1978 by CEB-FIP as

the “Model Code for Concrete Structure” as a model for the national codes to follow BS and IS codes have taken many of their Provisions from these publications

In the U.S.A., the Code of Practice published by the American Concrete Institute ACI 318-83, called Building Code Requirements for Reinforced Concrete, is currently used for design The philosophy of design used in this code is sometimes referred to as strength and serviceability design, and has the same basic ‘philosophy as the BS and IS ‘codes

4 LIMIT STATE DESIGN OF REINFORCED CONCRETE

1.6 DESIGN BY MODEL AND LOAD TESTS

prototypes In that case, instead of theoretical structural analysis of complicated structural combinations, tests are conducted on models made of materials like perspex of microconcrete Thus:

1 these tests can be used to give a very good physical idea of the action of these structures; or

2 the results of observations of deflections and strains interpreted by principles of model

analyses can be directly used for design; or

3 the results of the experimental model tests can be used to determine the boundary conditions and form the basis for complex computer analysis of the whole structure

The structural adequacy of reinforced concrete members which are factory made or precast in large quantities can also be tested for performance by means of laboratory tests on prototypes These tests give not only the strength but also the deflection and cracking performance of the structure under any given loading Many factory made products like prestressed concrete sleepers have been developed

by prototype testing

In those cases where the design and construction are to be finally passed on the basis of

concrete sleepers which will be ›subjected to a large number of repetitive loading during their life should be tested under millions of cyclic loads in addition to static tests

IS 456: clause 18.3 gives the following recommendations for designs based on experimental

1 The structure should satisfy the specified requirements for deflection and cracking when

In addition, there should be 75 per cent recovery of deflection after 24 hours of loading

collapse” for 24 hours

acceptance should be carried out according to IS 456: clause 16

1.7 “PUBLICATIONS BY BUREAU OF INDIAN STANDARDS

in India for limit state method-of design of R.C members The Bureau has also brought out the

related to

1.10

than the ¥

it (1978)

ctored load for

soth these tests

METHODS OF DESIGN OF CONCRETE STRUCTURES 5

By making use of these special publications, one will be able to design R.C structures with great speed and accuracy

REVIEW QUESTIONS 1.1 Enumerate the different methods of design of reinforced concrete members which are

accepted in practice

1.2 Name the codes of practice used for design of concrete structures for general building

purposes in (a) India, (b) U.K., (c) U.S.A., and (d) Germany

1.3 Give a short description of the following methods of design of reinforced concrete structures:

(a) Working stress methed

(b) Ultimate strength method

(c) Load factor method (d) Limit state.method (e) Strength and serviceability method

State the differences between the load factor method and the limit state method

1.4 What is meant by modular ratio? Why is it considered to be an unreliable quantity? What

is the difference in the value assumed for this quantity between IS and BS?

1.5 Explain the terms model and prototype of a structure

1.6 When will one use model studies for the design of a structure.as different from theoretical

calculations? How can model analysis- be used for design of concrete structures?

1.7 Explain the use of prototype testing in structural design Give examples as to where you

will recommend them

1.8 Give the IS specifications for load testing of prototypes for design based on experiments, stating the conditions to be satisfied Can field tests on a completed bridge be considered as

prototype testing? What are the loadings to be used for these acceptance tests?

1.9 What organisations are referred to as CEB and FIP and in what way is IS 456 (1978)

related to their publications?

1,10 Is the limit state method in any way a better method of design of concrete structures than the working stress design? Give reasons for your answer

1.11 Name the special publications by the Bureau of Indian Standards to supplement IS 456

(1978)

2 Partial Safety Factors in Limit State Design

2.1 INTRODUCTION

A structure is said to have reached its limit state, when the structure as a whole or in part becomes unfit for use, for one reason or another, during its expected life The limit state of a structure is the condition of its being not fit for use, and limit state design is a philosophy of design where one designs a structure so that it will not.reach any of the specified limit states during the expected life

of the structure

Many types of limit states or failure conditions can be specified The two major limit states

which are usually considered are the following:

1 The ultimate strength limit state, or the limit state of collapse, which deals with the strength and stability of the structure under the maximum overload it is expected to carry This implies that

no part or whole of the structure should fall apart under any combination of-expected overload

2 The serviceability limit state which deals with conditions such as deflection, cracking of

the structure under service loads, durability (under a given environment in which the structure has been placed), overall stability (i.e resistance-to collapse of the structure due to an accident such

as a gas explosion), excessive vibration, fire resistance, fatigue, etc

2.2 PRINCIPLES OF LIMIT STATE DESIGN Limit state design should ensure that the structure will be safe as regards the various limit state conditions, in its expected period of existence Hence the limit state method of design is also known

in American terminology as strength and serviceability method of design

The two major limit state conditions to be satisfied namely, the ultimate limit state and the serviceability limit state, are again classified into the following major limit states which are given

in the various clauses in 18 456 (1978)

Limit States

L

(i) Flexure (# 37) (i) Durability (# 7)

Gii) Shear (# 39) iii) Cracking ( 42)

[# refer.to clause in 1S 456 (1978)]

6

jor limit states

ith the strength

1is implies that

cted overload

mn, cracking of

ie structure has

| accident such

ious limit state

1is also known

it state and the

yhich are given

PARTIAL SAFETY FACTORS IN LIMIT STATE DESIGN 7

The usual practice of design of concrete structure by limit state principles consists in taking

up each of the above conditions and-providing for them separately so that the structure is safe under

all the limit states of strength and stability

2.3 PROCEDURE FOR DESIGN FOR LIMIT STATES

The design should provide for all the above limit state conditions;.each of these conditions is carried out as described now

1 Ultimate strength condition

The ultimate strength of the structure or member should allow an overload For this purpose, the structure should be designed by the accepted ultimate load theory to carry the specified overload

This may be in-flexure, compression, shear, torsion or tension

2 Durability condition The structure should be fit for its environment The cover for the steel as well as the cement content and water-cement ratio of the concrete that is provided in the structure should satisfy the given environmental conditions mentioned in Chapter 3

3 ‘Deflection condition

The deflection of the structure under service load condition should be within allowable limits This

@) Empirical method Since the most important empirical factor that controls deflection is

span/depth ratio, deflection can be controlled by limiting the span-depth ratios as specified by the

codes

(ii) Theoretical method Deflection can also be calculated by theoretical methods and controlled

by suitable dimensioning of the structure

(ii) Theoretical method The probable crack width is checked by theoretical calculations

5 Lateral stability against accidental horizontal loads (overall stability) This condition is met by observing the empirical rules given in codes for designing and detailing the vertical, horizontal, peripheral and internal ties in the structure

2.4 CHARACTERISTIC LOAD AND CHARACTERISTIC STRENGTHS Structures have to carry dead and live loads The maximum working load that the structure ‘has to

8 LIMIT STATE DESIGN OF REINFORCED CONCRETE

withstand and for which it is to be designed is called the characteristic load Thus there are characteristic

dead loads and characteristic live loads

The strengths that one can safely assume for the materials (steel and concrete) are called their characteristic strengths

For the sake of simplicity, it may be assumed that the variation of these loads ‘and strengths follows normal distribution law so that thé laws of statistics can be applied to them (see Fig 2.1)

Fig 2:1 Areas under the normal probability curve

As the design load should be more than the average load obtained from statistics (Fig 2.2),

we have

Characteristic design load = [Mean load] + K [Standard deviation for load]

As the design strength should be lower than the mean strength,

Characteristic strength = [Mean strength] — K [Standard deviation for strength}

The value of the constant K is taken by common consent as that corresponding to 5 per cent chance so that K will be equal to 1.64 as shown in Fig 2.1 (This is taken as 1.65 in Indian Standards.)

Even though the design load has to be calculated statistically as indicated above, research for

determining the actual loading on structures has not yet yielded adequate data to enable one to calculate theoretical values of variations for arriving at the actual loading on a structure Loads that

have been successfully used so far in the elastic design procedures are at present accepted as the characteristic loads The specified values to be used are laid down in IS 875

As stated earlier, the strengths that one can safely assume for steel and concrete are called their characteristic strengths Sufficient experimental data is already available about characteristic

strengths of steel and concrete These strengths are calculated from the theory of statistics and are

related to the standard deviation of the results of strength tests on the constituent materials

strength

to loads 2.5.1 The loa

symbol (Table

for load

has to b the ove! the chai

Structul

1 from th used fo

to be u

the vai the mồ

1 British

PARTIAL SAFETY FACTORS IN LIMIT STATE DESIGN 9

Fig 2.2 Characteristic strengths and characteristic loads

2.5 PARTIAL SAFETY FACTORS FOR LOADS AND MATERIAL STRENGTHS

Having obtained the characteristic loads and characteristic strengths, the design loads and design strengths are obtained by the concept of partial ‘safety ‘factors Partial safety factors are applied both

to loads on the structure and to strength of materials These factors are now explained

2.5.1 PARTIAL SAFETY FACTOR FOR LOAD % The load to be used for ultimate strengths design is also termed as factored load, In IS Code the

đ] i (Table 12 of IS 456 and Table 2.1 of the text) It may be noted that the use of partial safety factor

for load simply means that for calculation of the ultimate load for design, the characteristic load

has to be multiplied by a Partial safety factor denoted by the symbol y This may be regarded as

ics (Fig 2,2),

eth] the overload factor for which the structure has fo be designed Thus the load obtained by multiplying

the characteristic load by the partial safety factor is called the factored load, and is given by

65.in Indian : ì

Structures will have to be designed for this factored load

nable one to from that uséd-in elastic design It is the factored loads, and-not the characteristic loads, which are

to be used for calculating the factored loads as specified in IS 456 for various types of loads are given in Table 2.1

It may be noted that by adopting a partial safety factor of 1.5, both for dead and live loads,

Ị the value of the moment, shear force etc to be used in limit state design by IS Code is 1.5 times

te are called

aterials ũ Theoretically, the partial safety factors should be different for the two types of loads The

British Code BS 8110 uses a factor of 1.4 for DL and 1.6-for LL for strength considerations It is

10 LIMIT STATE DESIGN OF REINFORCED CONCRETE

TABLE 2.1 FACTORED LOADS FOR LIMIT STATE DESIGN

(Partial safety factors for loads)

(5.456: Table 12)

1 Dead and imposed 1.5 DL + 1.5 LL DL + LL

2 Dead and wind

Note: While considering earthquake effects, substitute EL for WL

only for convenience of using the same structural analysis for both elastic design and limit state design that IS recommends the same partial safety factor for dead:and live loads Thus in IS 456 the factored load, shear, moment etc in limit state design will be 1.5 times the value used for elastic design

2.5.2 PARTIAL SAFETY FACTORS FOR MATERIAL STRENGTHS ¥n

The grade strength of concrete is the characteristic strength of concrete, and the guaranteed yield strength of steel is the characteristic strength of steel Calculation’ {6 arrive at the characteristic

material strength of materials by using statistical theory takes into account only the variation of strength between the test specimens It should be clearly noted that the above procedure does not allow for the possible variation between the strength of the test specimen and the material in the structure which, as will-be seen in Section 2.6, is taken separately by a factor 0.67 The concept,

of partial safety factor for material strength due to variations in strength between samples is given

by the relation

Characteristic strength Design strength = Partial safety factor for strength, Ym This simply means that the strength to be used for design should be the reduced value of the characteristic strength by the factor denoted by the: partial safety factor for the material The

recommended values for these partial safety factors are given in Table 2.2;

TABLE 2.2 PARTIAL SAFETY FACTORS FOR STRENGTH, ¥,

dS 456: clause 35.4.2)

safety in elastic design as mentioned in Chapter 1 are usually 3 over the cube strength of concrete

for bending compression and 1.8 over the yield strength for steel stresses Thus in designing by working stress method, one works at stress levels well below the failure strength of concrete and

steel

It should also be remembered that the tables and formulae derived for limit state design and

those used in the Design Aids SP16 are derived with values of 7,, already incorporated in them

Hence, unlike the partial safety factor for load, these partial safety factors for material strengths

need not be considered.in routine design when using these formulae, charts and tables

2.6 STRESS-STRAIN CHARACTERISTICS OF CONCRETE

The mechanical properties of concrete, such as its stress-strain curve, depend on a number of

factors like rate of loading (creep), type of aggregate, strength of concrete, age of concrete, curing conditions, etc Figure 2.3a shows the typical stress-strain curves for concrete tested under standard conditions It can be seen that the failure strain is rather high and it is of the order of 0.4 per cent

when tested under constant strain rate

curves, and (b) Idealised curves

However, to derive an analytical expression for the stress-strain curve, it is necessary to idealise the curve By common consent, a rectangular parabolic curve (Fig 2.3b) has been accepted

as the stress-strain curve for concrete with the ultimate strain at failure as 0.0035, Codes differ with

respect to the strain €,, at which the strength becomes constant In IS it is taken as a constant value

of 0.002, and in BS, as a function of the strength of concrete and equal.to 2.4 x 1074 fal Ym + Thus,.the IS curve simplifies the distance at which the parabola ends and the rectangle begins Its

value can be deduced as follows: If x is taken as the depth of the neutral axis corresponding to the

42 LIMIT STATE DESIGN OF REINFORCED CONCRETE

strain 0.0035, the distance from the origin for the 0.002 strain is given by

x, = 02002

1ˆ 80035

=057x Thus the parabola extends to a distance (0.57x) and the rectangle for a distance 0.43x, as shown

in Fig 2.3 ˆ

The short term, static modulus for concrete, E, is assumed by IS code clause 5.2.3.1 as

E, = 5700 ff (N/mm?)

In most calculations this value has to be modified for creep and other long term effects

In order to distinguish between the concrete as tested in a cube and the concrete that exists

in the structure (size effect), it is assumed that the concrete in the structure develops a strength

of only 0.67 times the strength of the cube Hence the theoretical stress-strain curve of the

concrete in the design of structures is correspondingly reduced by the factor 0.67, as indicated in

Fig 2.3

In addition to the above and as explained earlier, a partial safety factor of 1.5 is applied on the concrete in the structure so that the design stress-strain curve for concrete in a structure will

be as shown in Fig 2.3 (IS 456; clause 37, Fig 20)

(distance of neutral axis)

2.7 STRESS-STRAIN CHARACTERISTICS OF STEEL

The stress-strain curve for steel.according to IS 456: clause 37.1 is assumed to depend on the type

of steel

Mild steel bar (f, = 250) is assumed to have a stress-strain curve as shown in Fig 2.4a and cold worked deformed bar (Fe 415) a stress-strain curve as shown in Fig 2.4b (Fig 22 of IS 456)

The stress-strain curves for steel, both in tension and compression in the structure, are assumed

to be the same as obtained in the tension test As the yield strength of IS grade steel has a minimum guaranteed yield strength, the partial safety factor to be used for steel strength need not be as large

as that for concrete The partial safety factor recommended for steel is 1.15, and this is to be applied

to the stress-strain curve as shown in Fig 2.4 (IS 456, Fig 22) It should be noted that for cold

worked deformed bars the factor 1.15 is applied to points on the stress-strain curve from 0.8f,

to f, only The value of E, is assumed as 200 kN/mm? for all types-of steels

In the revised BS 8110 (1985), the stress-strain curves for all steels used in reinforced concrete are simplified and assunied to be bilinear as in the case of mild steel bars The stress-strain curves for compression and tension are also assumed to be the same The curve used in BS 8110

is shown in Fig 2.5 l 2.8 SUMMARY OF DESIGN BY LIMIT STATE METHOD

The procedure to be followed in design by limit state method consists in examining the safety of the structure for at least all the important limit state conditions.explained in this chapter (strength,

durability, deflection cracking and overall stability)

Concepts of characteristic strengths and characteristic loads are used for design for strength

Separate partial safety factors for strength and loads are also introduced The design strengths

of steel and concrete are taken as the characteristic strength divided by their respective partial

for cont

computa

Th ~ Tension

Strain

Compression

Fig 2.5 Stress-strain curve for steel (BS 8110)

safety factors for strengths Similarly, the factored load to be resisted by the structure is taken as

the product of the characteristic load and the partial safety factor for loads The stress-strain curve

for concrete and steel are assumed to be of fixed shape, for convenience in mathematical

14 LIMIT STATE DESIGN OF REINFORCED CONCRETE EXAMPLE 2.1 (Calculation of factored or design loads)

A one-way slab for a public building (loading class 300) is 200 mm in overall thickness It is simply supported on a span of 4 m Determine the factored moment and factored shear for strength design and the loads for checking serviceability

Factored (design) moment M„

EXAMPLE 2.2 (Calculation of factored loads)

A column 4 m high is fixed at the base and the top end is free It is subjected to the following loads:

Total DL = 40.kN

Total imposed (gravity) load = 100 KN

Wind load = 4 KN per metre height

Determine the factored loads (a) for strength, and (b) serviceability limit states

1 Total wind load : (Loads in KN) :

EXAMPLE 2.2 (cont.)

Ref Step Calculations Output

Table 12 = 210 KN (Vertical) P=210

Gi) Dead + wind (Dead load assists overturning)

Vertical = 0.9 DL = 0.9 x 40 = 36 KN P = 36 Horizontal = 1.5 WL = 1.5 x 16 = 24 kN H=24

Gii) Dead + imposed + wind Vertical = 1.2 (DL + LL) = 1.2 (40 + 100)

3 Loading for serviceability design

Gi) Dead + wind = 1.0 (DL + WL)

EXAMPLE 2.3 ‘(Calculation of design loads)

An R.C column of 500 mm_dia has to carry a direct load of 900 KN and a moment of 100 kNm about the YY-axis due to characteristic dead and live loads A seismic moment of 500 kNm is estimated to be felt on the column in any direction Calculate the design loads for the pile and pile caps for a layout of the piles, as shown in Fig E.2.3

Ref Step Calculations Output

16 LIMIT STATE DESIGN OF REINFORCED CONCRETE

EXAMPLE 2.3 (cont.) Ref Step Calculations Output

(Say, upwards on D and C and downwards on A and B)

Load on each pile = 45.45 kN

3 Load due to earthquake moment (EL)

= ag = 4545 kN

(upwards or downwards)

Load on each pile = 227.25 kN (Loads in KN)

4 Maximum load on the pile due to VL + ML 1S 456 = 1.5 (225 + 45.45) = 405.7 kN (max) P=405.7

2.2 Enumerate the five limit states commonly used-in limit state design and state briefly how they are provided for in design :

2.3 Explain how 5 limit state design is very similar to the strength and serviceability design

of the American Code

2.4 What is meant by characteristic strength of a material as used in IS 456 (1978)?

2.5 What is meant by normal distribution in statistics and what is the relationship between mean value and characteristic value in such a distribution assuming 5 per cent confidence limit?

2.6 Define the term ‘partial safety factors’ as used in limit state design Identify the various factors and state the values recommended in IS 456

2.7 Explain the terms ‘factored load’ and ‘characteristic loads’ Why does IS 456 specify the

recommended in BS code? ` 2.8 Distinguish between the terms ‘factor of safety'-and ‘partial safety factor’ for material strength What are the usual factors of safety used in elastic design of R.C members?